"What I am wondering about these days is when are children developmentally ready to understand the concept of ratios and relationships? Personally, I love having been able to memorize all my “math facts,” but I thought, really really believed that doing calculations was doing math. If I had understood that math was more about the discovery and description of relationships it would have made a difference. I think that this is where the Wall might be: when students need to shift from “getting an answer” to understanding relationships. The calculations mind-set is so deeply hard-wired into the brain that this shift never happens.
So I am wondering, from your work with young children, when do you think you can start talking about math as if it’s tool for discovery?"

Paula puts relationships and ratios together; you could include proportion and scaling. You could include fractions or percentages. It's all part of one concept. I chose scaling as the word we would use, as it seems like the most everyday way of talking about it. We talk about scale models, the scale of a map, scaling up a recipe or picture. I dropped in the words "ratio" and "proportion" occasionally, but not in any way I wanted the kids to remember. It was about the experiences primarily.

Reflecting on the lessons made me think a little more about what scaling is. It's not that easy to encapsulate what it's essentially about, and I'd appreciate other people's views on this. Thinking through what's common to all the activities, it seems it's a kind of "I'm taking you with me" thing. Change one thing, other things have to change too.

It's much easier for me to think about in particular examples. The longer the car journey is, the bigger other things get. You need a bigger bag of travel sweets for instance.

So, how well did these lessons do? (Click on the numbers in circles for more detail on each activity.)

➊Scaling up shapes on grid paper. Here, what goes together is the horizontal and the vertical scaling. Some of the children at first scaled one way but not both. But in the end they all got it. It's a really satisfying lesson for the children this, because they're creating their own shape, and the challenge level is just right.

Of course, there's something really interesting going on with the area within these shapes - there are square numbers involved - but though we touched on this, the thrust of the lesson was just to be able to create the shapes.

Julie, did something good with it in the other Year 4 class: she got the kids to add numbers to label all the lengths. This makes the numerical patterns much more obvious, and I'd do that next time.

There was triangle grid paper and some of the children wanted to use that. I went with it because I wanted to see how they'd scale up the three dimensions. But it's such a wonderful thing being able to draw 3D things on flat paper that some of them got a bit carried away with the first shape, and couldn't really scale it up easily!

If I had the same children again in later years, I would start to look at the areas and the volumes involved, how if you graph them, you don't get a straight line. This kind of non-linear scaling is really important in science, engineering and industry, not to mention maths itself!

➋Using the pattern blocks is a nice easy activity, and kids a few years younger could do this, making sure there were twice the number of triangles as squares. Because it was easy I could use it to link this scaling to its graphical representation as a diagonal line.

❸ Folding and halving A-size paperis something younger children could do. It could also go off in the directions of fractions and even the idea of infinite series. (I didn't mention in the post how we looked at other rectangles that don't scale up this way - and got into an interesting discussion about squares in particular.) We touched on the fact that the A-sizes have a 1:√2 ratio, but this is something older year groups could explore more.

❹The 1:100 and 1:500 scale models of the A380 came along at just the right time. Children have so much experience with scale when it comes to models. And as we were looking at maps of runways for our work on headings, it was natural to ask, if this is the plane scaled down, how much would the runway scaled down be? We could have done more on this if time allowed, and scales in maps is such an obvious place to deal with scaling.

➎Cuisenaire rods gave us the chance to pass by lesson 1 again. I think younger children could do this too. What's special about this approach is that it makes the square numbers really apparent, so it would help older kids too. You can actually stand blocks on their end in the trays, so you could scale up in three dimensions too, and think about cube numbers.

➏Kids should cook regularly anyway, but this time we especially focused on the scaling up the recipe. (I had to contrive it a bit, because the recipe was right to start with!) The online activity was good too. It showed me that scaling from a recipe for 2 to a recipe for 3 was very challenging for them. That's where to go next year and beyond. This activity also has the advantage that you can eat it!

➐The Zoolander question from Robert Kaplinksy was, in hindsight, quite challenging for this age group, maybe too challenging. The class got the humour of Zoolander not understanding a scale model though!

❦

Strangely perhaps, I'm not satisfied with all this. And I don't even know why. Is it that scaling is such a slippery concept? Is it that the work we did didn't involve much calculation? Is it just that we need to be doing things like this every year? Maybe it's this... I don't know.

I'd be interested in your ideas about this. What, for you, is the essence of scaling? Do you think it's worth devoting time to? Are there other ways of approaching it you would choose? Let me know!

The idea is, we teach arithmetic, sure, but what about noticing the patterns in what we do, making generalisations, or claims, about the ways numbers behave?

It's not about using algebraic notation - far from it - it's about noticing patterns.

An example from our Y4 classes this year: we asked children to look at patterns if you add consecutive numbers. Quite a few children noticed and made the claim that you always get an odd number if you add two consecutive numbers. Some of them could justify the claim. We'd already had a lesson about the addition of odd and even numbers, so they could refer back to that. Eventually we had a whole load of claims, which I tweeted:

There's more going on here too. It's not just about meta-thinking as subject matter, but the emphasis of the book - and often my approach - is the way this thinking is done. Here's an excerpt of one of the many conversations in the book:

Ms. Diaz: ... Have you ever spent time thinking about how you participate in discussions? Like what do you do when something is hard for you to think about? Or when you don't get what someone is saying?

Kathryn: Well, no one has ever asked me to think about this before. Usually, it is like we just have to have silence.

Ms Diaz: When I was in school, we didn't spend time talking. Only the teacher did the thinking. But I want us to be a team so that we can all contribute.

Will: I like how we show our thinking, you know like we come up to the overhead and show our thinking. I like how lots of people have different ideas.

Brent: It is like there are lots of teachers.

What do you see in this snatch of conversation?

I'm reminded of Freemont's four freedoms mentioned in my last post: freedom to make mistakes, to think for yourself, to ask questions and to choose a method of solution. Conversations like this are places to find these freedoms. Where the teacher asks questions, and genuinely wants to hear what the students think about the question, rather than something that they've told them to learn. It's worth reading Ilana Horn's post on Asking the Right Question.

How do we do that in math class, the place with the most deep-seated rituals of recitation and mindless calculation? How do we move from what English mathematician and philosopher Alfred North Whitehead called ‘inert ideas’ — those which have been received and not utilized or tested or ‘thrown into fresh combinations’?

(I should add, I also love Ralph Pantozzi's coin-tossing activity in the video in the last post, even though, it's not about reflective conversation. It's a teacher-orchestrated activity, and I do plenty of those. One like this can clearly be really enjoyable and memorable and can lead on to opportunities for reflection and conversation. Nicole Louie however mentions "good task worship" in her comment on Ilana's post, and I've been thinking about that lots. You see, I'm always looking for good tasks. Am I a worshiper? I'm still pondering that...)